MCQ
Write the correct answer in the following: A diagonal of a rectangle is inclined to one side of the rectangle at $25^\circ$. The acute angle between the diagonals is:
- A$55^\circ$
- ✓$50^\circ$
- C$40^\circ$
- D$25^\circ$
✓
Answer
Correct option: B.
$50^\circ$
We know that, digonals of a rectangle are equal in length.
$\therefore\ \text{AD}=\text{BD}$
$\Rightarrow\ \frac{1}{2}\text{AC}=\frac{1}{2}\text{BD}$ [dividing both sides by $2$]
$\Rightarrow\ \text{OA}=\text{OB}$ [since, $O$ is the mid-point of $AC$ and $BD$]
$\Rightarrow\ \angle2=\angle1$ [angles opposite to equal sides are equal]
$=25^\circ$
$\therefore\ \angle3=\angle1+\angle2$
[exterior angle is equal to the sum of two opposite intersite interior angles]
$=25^\circ+25^\circ=50^\circ$
Hence, the acute angle between the diagonals is $50^\circ$.
$\therefore\ \text{AD}=\text{BD}$
$\Rightarrow\ \frac{1}{2}\text{AC}=\frac{1}{2}\text{BD}$ [dividing both sides by $2$]
$\Rightarrow\ \text{OA}=\text{OB}$ [since, $O$ is the mid-point of $AC$ and $BD$]
$\Rightarrow\ \angle2=\angle1$ [angles opposite to equal sides are equal]
$=25^\circ$
$\therefore\ \angle3=\angle1+\angle2$
[exterior angle is equal to the sum of two opposite intersite interior angles]
$=25^\circ+25^\circ=50^\circ$
Hence, the acute angle between the diagonals is $50^\circ$.
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